A226939 A recursive variation of the Collatz-Fibonacci sequence: a(n) = 1 + min(a(C(n)),a(C(C(n)))) where C(n) = A006370(n), the Collatz map.

1, 1, 4, 2, 3, 5, 9, 2, 10, 4, 8, 5, 5, 9, 9, 3, 7, 11, 11, 4, 4, 8, 8, 6, 12, 6, 56, 10, 10, 10, 54, 3, 14, 7, 7, 11, 11, 11, 18, 5, 55, 5, 15, 9, 9, 9, 53, 6, 13, 13, 13, 6, 6, 57, 57, 10, 17, 10, 17, 10, 10, 54, 54, 4, 14, 14, 14, 8, 8, 8

Offset: 1

Author

Andres M. Torres, Jun 22 2013

Keywords

• nonn

Comments

The sequence contains mysterious duplicates of terms, sometimes in groups of 2 to 4 at a time, but I haven't seen any cyclic patterns, it's all unique.

Examples

			a(n) values frequently depend on both lesser and higher terms:
a(3)= 1+ min( a(C(3)), a(C(C(3)))) = 4
a(3)= 1+ min( a(10), a(5))= 1+min(4,3) = 4
a(10)=1+ min( a(5), a(16))= 1+min(3,3) = 4
a(5) =1+ min( a(16),a(8)) = 1+min(3,2) = 3
a(16)=1+ min( a(8), a(4)) = 1+min(2,2) = 3
a(8) =1+ min( a(4), a(2)) = 1+min(1,1) = 2
a(4) =1+ min( a(2), a(1)) = 1+min(1,1) = 2
a(2) =1 (starting value)
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014682.

Programs

Blitz3D

function A(n)
            if n=1 or 2
                   return 1
            else
                   return 1 +lesser(A(C(n)), A(C(C(n))))
            end if
end function
; The Collatz Sequence generator equation
Function C(n)
       If n Mod 2
                 Return 3*n+1
       Else
                 Return n Shr 1
       End If
End Function
;; Andres M. Torres, Jun 26 2013

PARI

C(n)=if(n%2,3*n+1,n/2)
A=vector(10^4);A[1]=A[2]=1;
a(n)=if(n<=#A && A[n], A[n], my(c=C(n),t=min(a(c), a(C(c)))+1); if(n>#A, t, A[n]=t)) \\ Charles R Greathouse IV, Jun 24 2013

Formula

a(n) = 1 + min(a(C(n)), a(C(C(n)))), where C(n) = A006370(n).